Find x: A Simple Formula to Solve x + x^2 + x^3 +...+ x^n = 1

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The equation x + x^2 + x^3 + ... + x^n = 1 can be solved using the geometric series identity, reformulating it to x(x^n - 1)/(x - 1). This leads to the polynomial equation x^(n+1) - 2x + 1 = 0. While this transformation simplifies the problem, there is no known general solution for arbitrary values of n. This discussion is particularly relevant for those in accounting or MBA fields seeking mathematical solutions.

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I need to calculate easy solution for following equation.

x + x^2 + x^3 +...+ x^n = 1

I need a simple formula to calculate value of x that satisfies above equation for any value of n.

This is for accounting/MBA has nothing to do with Physics, but I am sure anyone will be able to help me out.

Thanks.
 
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Using the geometric series identity: [itex]x+x^{2}+...+x^{n}=x\frac{x^{n}-1}{x-1}[/itex]
we may reformulate your equation to:
[tex]x^{n+1}-2x+1=0[/tex]
I'm not too sure there exist a nice, general solution of this for arbitrary n.
 

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